Bulletin T.CXXXIV de l’Académie serbe des sciences et des arts − 2007
Classe des Sciences mathématiques et naturelles
Sciences mathématiques, No 32
ESTRADA INDEX OF ITERATED LINE GRAPHS
TATJANA ALEKSIĆ, I. GUTMAN, M. PETROVIĆ
(Presented at the 2nd Meeting, held on March 23, 2007)
A b s t r a c t. If λ1 , λ2 , . . . , λn are the eigenvalues of a graph G , then the
Estrada index of G is EE(G) =
n
P
eλi . If L(G) = L1 (G) is the line graph
i=1
of G , then the iterated line graphs of G are defined as Lk (G) = L(Lk−1 (G))
for k = 2, 3, . . . . Let G be a regular graph of order n and degree r . We show
that EE(Lk (G)) = ak (r) EE(G) + n bk (r) , where the multipliers ak (r) and
bk (r) depend only on the parameters r and k . The main properties of ak (r)
and bk (r) are established.
AMS Mathematics Subject Classification (2000): 05C50
Key Words: spectrum (of graph), Estrada index (of graph), regular
graph, line graph, complex networks
1. Introduction
Let G be a graph of order n . The eigenvalues of the adjacency matrix of
G , denoted by λ1 , λ2 , . . . , λn , are said to be the eigenvalues of G , and they
form the spectrum of G [3]. As well known [3], the spectra of graphs have
found numerous applications in various fields of science. One of the newest
such application was put forward by Ernesto Estrada, who arrived at the
conclusion that the graph–spectrum–based graph invariant
EE = EE(G) =
n
X
i=1
eλi
(1)
34
Tatjana Aleksić, I. Gutman, M. Petrović
can be used as a measure of the degree of folding of long–chain polymeric
molecules, especially those of biological importance [5, 6, 7]. Somewhat later,
Estrada and Rodrı́guez–Velázquez used the same quantity EE to describe
certain properties (so-called “centrality”) of complex networks [9, 10]; some
most recent results along these lines can be found in [8]. In what follows,
EE will be called the Estrada index .
The study of the mathematical properties of the Estrada index started
only quite recently [4, 11, 12], and relatively little is known on its dependence
on the structure of the underlying graph. In this work we report on the
properties of the Estrada index of iterated line graphs of regular graphs.
Let, as usual, L(G) denote the line graph of the graph G . For k =
1, 2, . . . , the k-th iterated line graph of G is defined as Lk (G) = L(Lk−1 (G)) ,
where L0 (G) = G and L1 (G) = L(G) .
If G is a regular graph of degree r = 0 , then Lk (G) , k ≥ 1 , are graphs
without vertices. In this case, EE(G) = n and EE(Lk (G)) = 0 for k ≥ 1 .
If G is a regular graph of degree r = 1 , then L(G) consists of isolated
vertices, and Lk (G) , k ≥ 2 , are graphs without vertices. In this case,
EE(G) = (n/2)(e+1/e) , EE(L(G)) = n/2 , and EE(Lk (G)) = 0 for k ≥ 2 .
If G is a regular graph of degree r = 2 , then L(G) ∼
= G and, consequently,
k
k
∼
L (G) = G for all k ≥ 1 . In this case, EE(L (G)) = EE(G) for all k ≥ 1 .
Therefore, in order to avoid these trivial cases, in what follows we assume
that the degree r of the regular graphs considered is at least three.
The line graph of a regular graph G of order n = n0 and of degree
r = r0 is a regular graph of order n1 = nr/2 and of degree r1 = 2r − 2 .
Consequently, all the iterated line graphs of a regular graph are regular
graphs. In particular, the order nk and degree rk of Lk (G) , k ≥ 1 , obey
the recurrence relations
nk =
1
nk−1 rk−1
2
;
rk = 2 rk−1 − 2
(2)
from which one directly obtains [1, 2]:
rk = 2k r − 2k+1 + 2
nk =
´
Y
Y³
n k−1
n k−1
2i r − 2i+1 + 2 .
r
=
i
k
k
2 i=0
2 i=0
(3)
(4)
35
Estrada index of iterated line graphs
For the sake of simplicity we define


1




γk (r) :=
for k = 0
(5)
´
Y³

1 k−1

i
i+1

2
r
−
2
+
2

 2k
for k ≥ 1
i=0
and then Eq. (4) can be rewritten as
nk = γk (r) n .
(6)
Both Eqs. (3) and (6) hold for k ≥ 0 .
According to a classical result of Sachs [13], if λ1 , λ2 , . . . , λn are the eigenvalues of a regular graph G of order n and of degree r , then the eigenvalues
of its line graph L(G) are
λi + r − 2
i = 1, 2, . . . , n
−2
n(r − 2)/2

and 

times .
(7)
Combining formulas (1) and (7) one immediately arrives at
Lemma 1. If G is a regular graph of order n and degree r ≥ 3 , then
the Estrada indices of G and L(G) are related as
EE(L(G)) = er−2 EE(G) +
r−2
n.
2 e2
(8)
Lemma 2. If G = L0 (G) is a regular graph of order n = n0 and degree
r = r0 ≥ 3 , then for k ≥ 0 , the Estrada indices of Lk (G) and Lk+1 (G) are
related as
EE(Lk+1 (G)) = erk −2 EE(Lk (G)) +
rk − 2
nk .
2 e2
(9)
Evidently, for k = 0 , Eq. (9) reduces to Eq. (8). It should be noted
that Eqs. (8) and (9) hold also in the cases r = 0, 1, 2 .
By a two-fold application of Lemma 2 we get
EE(L2 (G)) = er1 −2 EE(L1 (G)) +
r1 − 2
n1
2 e2
·
= er1 −2 er0 −2 EE(L0 (G)) +
¸
r0 − 2
r1 − 2
n0 +
n1 ,
2 e2
2 e2
36
Tatjana Aleksić, I. Gutman, M. Petrović
i.e.,
EE(L2 (G)) = er1 −2 er0 −2 EE(L0 (G)) + er1 −2
r0 − 2
r1 − 2
n0 +
n1 . (10)
2
2e
2 e2
In a similar manner,
EE(L3 (G)) = er2 −2 er1 −2 er0 −2 EE(L0 (G)) + er2 −2 er1 −2
r0 − 2
n0
2 e2
r1 − 2
r2 − 2
n1 +
n2
2 e2
2 e2
+ er2 −2
(11)
EE(L4 (G)) = er3 −1 er2 −2 er1 −2 er0 −2 EE(L0 (G)) + er3 −2 er2 −2 er1 −2
+ er3 −2 er2 −2
r0 − 2
n0
2 e2
r1 − 2
r2 − 2
r3 − 2
n1 + er3 −2
n2 +
n3 .
2
2
2e
2e
2 e2
(12)
It is not difficult to see that Eqs. (10)–(12) are special cases of the formula:
EE(Lk+1 (G)) =
à k
Y
!
eri −2
EE(G)+
k−1
X
p=0
i=0


k
Y
i=p+1

eri −2 
rp − 2
rk − 2
np +
nk .
2
2e
2 e2
(13)
2. The Main Result
According to Eq. (3), the terms r0 , r1 , . . . , rk in the expression
k
Y
eri −2
(14)
i=0
occurring on the right–hand side of (13) are fully determined by the degree
r of the graph G . Thus, (14) depends solely on r and k .
According to Eqs. (4) and (6), the terms n0 , n1 , . . . , nk in the expression
k−1
X
p=0


k
Y
i=p+1

eri −2 
rp − 2
rk − 2
np +
nk
2 e2
2 e2
(15)
occurring on the right–hand side of (13) are fully determined by the order
n and degree r of the graph G and are, in addition, linearly proportional to
37
Estrada index of iterated line graphs
the parameter n . Consequently, (15) depends solely on n , r , and k , and is
linearly proportional to n .
Bearing the above in mind we arrive at:
Theorem 3. The Estrada index of a regular graph G of order n and
degree r ≥ 3 , and of its iterated line graphs are related in the following
manner
EE(Lk (G)) = ak (r) EE(G) + bk (r) n
where ak (r) and bk (r) are functions depending solely on the variable r and
parameter k , k ≥ 1 .
3. On the Functions ak (r) and bk (r)
From (8) we immediately see that
a1 (r) = er−2
r−2
.
2 e2
b1 (r) =
Using the relations (2), from (10) we obtain
a2 (r) = e3(r−2)
b2 (r) =
i
r − 2 h 2(r−2)
e
+
r
.
2 e2
Analogously, from (11) there follows
a3 (r) = e7(r−2)
b3 (r) =
i
r − 2 h 6(r−2)
e
+ r e4(r−2) + 2r(r − 1) .
2
2e
Using the relation (3), the general form of the function ak (r) is easy to
find. For k ≥ 1 , from (14),
ak (r) =
k−1
Y
ri −2
e
i=0
which finally yields
=
k−1
Y
e(
2i
)−2 =
r−2i+1 +2
i=0
k−1
Y
P
k−1
e
(r−2) 2i
= e i=0
(r−2) 2i
,
i=0
k −1)
ak (r) = e(r−2)(2
.
(16)
38
Tatjana Aleksić, I. Gutman, M. Petrović
The function bk (r) cannot be transformed into a similarly simple form.
For k = 1, 2, 3 , expressions for bk (r) are given above. Therefore, in what
follows we assume that k ≥ 4 . By combining (6) and (15) we get
bk (r) =
k−2
X
p=0


k−1
Y

eri −2 
i=p+1
rp − 2
rk−1 − 2
γp +
γk−1 ,
2 e2
2 e2
which after a lengthy calculation, taking into account (5), becomes
bk (r) =
r − 2 (r−2) 2k
e
×
2 e2

× e−2(r−2) + r e−4(r−2) +
k−2
X
re
−(r−2)2p+1
2p−1
p=2
+
p−1
Y³
´

2i−1 r − 2i + 1 
i=1
´
Y³
r(r − 2) k k−2
i−1
i
2
2
r
−
2
+
1
.
8 e2
i=1
Both ak (r) and bk (r) are rapidly increasing functions of both r and k .
From Eq. (16) it is immediately seen that for k → ∞ ,
³
k
ak (r) = O e(r−2) 2
´
i.e., that for all values of r , 0 < r < ∞ ,
ak (r)
(r−2)
2k
k→∞ e
0 < lim
<∞.
We now show that the asymptotic behavior of the function bk (r) is analogous, namely that
³
´
k
bk (r) = O e(r−2) 2
.
(17)
In view of Eq. (4), the term [(rk − 2)/(2 e2 )] nk , occurring in Eq. (9),
can be written as K dk , where
K=
1
nr(r − 2) e−2 = const
4
and
dk := 2k
k−1
Yh
i
2i−1 (r − 2) + 1 .
i=1
(18)
39
Estrada index of iterated line graphs
Then
EE(Lk+1 (G)) = erk −2 erk−1 −2 · · · er2 −2 EE(L2 (G)) + ck ,
where
ck := K
k
X
erk −2 · · · erp −2 dp−1 + K dk
p=3
(r−2) 2k+1
=
Ke
k
X
e
−(r−2) 2p
p−1
2
p=3
p−2
Yh
i
2i−1 (r − 2) + 1 + K dk .
i=1
The sequence {fk } , defined via
fk :=
k
X
p
e−(r−2) 2 2p−1
p=3
p−2
Yh
i
2i−1 (r − 2) + 1
i=1
is bounded from above. To see this note that for r > 2 and i ≥ 1 ,
(r − 2) 2i−1 + 1 ≤ (r − 2) 2i−1 + (r − 2) 2i−1 = (r − 2) 2i
from which
p−2
Yh
i−1
(r − 2) 2
i
+1
≤
i=1
p−2
Y
(r − 2) 2i = (r − 2)p−2 21+2+···+(p−2)
i=1
= (r − 2)p−2 2(p−1)(p−2)/2
from which
fk
=
k
X
p
e−(r−2) 2 2p−1
p=3
leq
k
X
p−2
Yh
i
2i−1 (r − 2) + 1
i=1
p
e−(r−2) 2 2p−1 (r − 2)p−2 2(p−1)(p−2)/2
p=3
=
k
X
(r − 2)p−2 2p(p−1)/2
p=3
e(r−2) 2p
<
+∞
X
(r − 2)p−2 2p(p−1)/2
.
e(r−2) 2p
p=3
40
Tatjana Aleksić, I. Gutman, M. Petrović
The latter series converges by D’Alembert’s criterion. Indeed, let
gp :=
Then the quotient
(r − 2)p−2 2p(p−1)/2
.
e(r−2) 2p
gp+1
(r − 2) 2p
= (r−2) 2p
gp
e
evidently tends to zero for p → ∞ .
Since the series {gp } converges, the sequence {fk } is bounded. Therefore,
for k → ∞ ,
ck − K dk = K e(r−2) 2
k+1
³
k+1
fk = O e(r−2) 2
´
.
Using the above specified notation,
k+1
bk+1 (r) = hk + K e(r−2) 2
where
k+1 −4)
hk := e(r−2)(2
·
fk + K dk
(19)
r − 2 2(r−2) r(r − 2)
e
+
2 e2
2 e2
For k → ∞ ,
³
hk = O e(r−2) 2
k+1
¸
.
´
and, consequently,
k+1
hk + K e(r−2) 2
³
k+1
fk = O e(r−2) 2
´
.
(20)
Let p be the smallest integer for which r − 2 ≤ 2p holds. Then from (18)
there follows
dk < 2k
k−1
Yh
i
2i (r − 2) ≤ 2k
i=1
k−1
Yh
i
2i 2p = 2p(k−1)+k(k+1)/2
i=1
for r > 2 , which implies
dk
(r−2)
2k+1
k→∞ e
lim
=0.
Therefore, for k → ∞ ,
³
k+1
dk = o e(r−2) 2
´
³
⇒
dk = O e(r−2) 2
k+1
´
.
(21)
Estrada index of iterated line graphs
41
The property (17) of the function bk (r) follows now when the relations (20)
and (21) are combined with (19).
Acknowledgement. This work was also supported by the Serbian Ministry
of Science and Environmental Protection, through Grant no. 144015G.
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Faculty of Science
University of Kragujevac
P. O. Box 60
34000 Kragujevac
Serbia
taleksic@kg.ac.yu
gutman@kg.ac.yu
petrovic@kg.ac.yu